Return again to the equation (REF ), which is a simplified equation (REF ).
(a) Compute the scale function \(s\)  for the diffusion \(C_k\) : Its derivative is
\(s^{\prime }(c) = \exp \left[-2\int _0^c\frac{\tilde{\gamma }(u)}{\tilde{\sigma }^2(u)}\,\mathrm {d}u\right].\) 
There exist \(\gamma _* > 0\)  and \(c_* > 0\)  such that
\(\tilde{\gamma }(u) \le -\gamma _*,\, c \ge c_*;\quad \tilde{\gamma }(u) \ge \gamma _*,\, c \le -c_*.\) 
Moreover, \(\tilde{\sigma }(u) \ge \sigma (u) \ge \sigma _*\)  for all \(u \in \mathbb {R}\) . Thus for \(c \ge c_*\) ,
\(s^{\prime }(c) \ge s^{\prime }(c_*)\exp \left[2\int _{c_*}^c\frac{\gamma _*}{\sigma _*^2}\,\mathrm {d}c\right] = s^{\prime }(c_*)\exp \left[2(c-c_*)\frac{\gamma _*}{\sigma _*^2}\right].\) 
A similar estimate is true for \(c \le -c_*\) :
\(s^{\prime }(c) \le s^{\prime }(-c_*)\exp \left[2(|c|-c_*)\frac{\gamma _*}{\sigma _*^2}\right].\) 
The speed measure has bounded Lebesgue density \(1/(s^{\prime }(c)\sigma ^2(c))\) , as shown in (REF ), (REF ), (REF ). Apply Feller's test from [1]} to complete the proof.
(b) The statement for \(\pi _C\)  follows from ellipticity of the elliptic partial differential equation governing the Lebesgue density of this stationary distribution. The statement for \(\pi _{\mu }\)  follows from one-to-one mapping between \(\mathbf {C}\)  and \(\mu \) .
